I suppose that "non-compact complex algabraic curve" means complex affine curve.
Then the answer is negative. The simplest example is this: let $C=D=$ triply punctured sphere, and degree is $6$. (There is only one affine curve isomorphic to the triply punctured sphere). One can construct two rational functions $p_1,p_2$ of degree $6$, each with three critical points and three critical values, such that the local degrees of $p_1$ at the critical points are $(3,5,5)$ and local degrees of $p_2$ are $(4,4,5)$. Evidently there is no isomorphism $\phi$ such that $p_1=p_2\circ\phi$.
Various constructions of such $p_1,p_2$ are available, see, for example
A. Eremenko, A. Gabrielov, M. Shapiro and A. Vainshtein, Rational functions and real Schubert calculus, Proc. AMS, 134, 4 (2005) 949-957.
Alternatively, one can use subgroups of the modular group. Or designs d'enfant:
L. Schneps, Dessins d'enfants on the Riemann sphere. The Grothendieck theory of dessins d'enfants (Luminy, 1993), 47–77, London Math. Soc. Lecture Note Ser., 200, Cambridge Univ. Press, Cambridge, 1994.